I have experienced the same behaviour. However, I did manage to get the score to look how I wanted it, which is the same as it would appear within one single flow if there was a change of meter/key at a system break (ie cautionary time- and key-sig’s.).
I set the final barline of the flow to be “no barline” (Notation Options > Barlines > Automatic barline at end of flow: > No barline) and used Edit > Remove Rests to delete the rest in the bar that Dorico adds. This appearance in the score was not the same in the parts, though. Every part still had that obstinate bar rest at the end of the flow, albeit with no final barline. No matter what I tried, I could not remove the rest from the bar in the parts.
The problematic behaviour was the same irrespective of whether it was with a cautionary time- or key-sig by itself or both together.
Curiously, the problem in the parts only manifested itself when Layout Options > Players > Bar Rests and Multi-bar Rests > Consolidate: was set to Multi-bar Rests. When it was set to None, the “rogue” bar disappeared from the parts. Unfortunately, all of the parts had quite a number of places requiring the use of multi-bar rests.
I think that it is (currently) not possible to have a cautionary key signature without something following it immediately within the same flow, as what we call the cautionary key-sig is treated by Dorico as advance warning of an upcoming key-sig which actually does exist within that flow.
In my case, as the choral/orchestral piece was only about seven minutes long, I ended up combining the four flows into one. The cautionary time and key signatures then behaved as expected in both the score and the parts.